The conjugate momenta \(E=-P_t\) , \(P_z\) , \(J_\phi = P_\phi \) , and \(J_\psi =P_\psi \)  are conserved but the spherical symmetry \(SO(3)\times SO(3)\)  of the small BH is broken to \(SO(2)_\phi \times SO(2)_\psi \)  by the very presence of the circular profile.
Despite the reduced isometry, very much as for Kerr BHs, separation is possible à la Carter [1]}, [2]}:
\(&P_\rho ^2=Q_R(\rho )=\mathcal {E}^2\left(1+{L_1^2+L_5^2\over \rho ^2+a_f^2}\right)+{\mathcal {L}_\phi ^2\over (\rho ^2+a_f^2)^2}-{\mathcal {L}_\psi ^2\over \rho ^2(\rho ^2+a_f^2)}-{K^2\over \rho ^2+a_f^2}\\\nonumber &P_\theta ^2=Q_A(\theta )=K^2-{J_\psi ^2\over \cos ^2\theta }-{J_\phi ^2\over \sin ^2\theta }-\mathcal {E}^2a_f^2\sin ^2\theta \) 
